Large Gauge Symmetry and Memory Effects in Electron-Ion ... · Large Gauge Symmetry and Memory...

Large Gauge Symmetry and Memory Effectsin Electron-Ion Collisions

Monica Pate

Harvard University

September 12, 2018

Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 1 / 28

Key Points

1. Generic gauge theories enjoy infinite-dimensional physical symmetries,known as asymptotic or large gauge symmetries.

2. The memory effect is the observable consequence of large gaugesymmetry.? The memory effect is the effect of a vacuum transition on a pair of probes

charged under the gauge group.

3. The color memory effect appears in scattering at collider energies in theRegge limit of QCD.

4. In scattering events, large gauge symmetry implies soft radiation (orvacuum transitions) is highly correlated with hard particles.


Key Points

1. Generic gauge theories enjoy infinite-dimensional physical symmetries,known as asymptotic or large gauge symmetries.? In systems with asymptotic regions or boundaries, part of the gauge

symmetry may be physical (as opposed to redundant).? The physical symmetries have physical consequences.? Canonical example is the conservation law derived by Noether’s theorem.? Noether has a second theorem for local symmetries.

2. The memory effect is the observable consequence of large gaugesymmetry.? The memory effect is the effect of a vacuum transition on a pair of probes

charged under the gauge group.3. The color memory effect appears in scattering at collider energies in the

Regge limit of QCD.4. In scattering events, large gauge symmetry implies soft radiation (or

vacuum transitions) is highly correlated with hard particles.Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 3 / 28

Noether’s First Theorem

I The variation of the action under a symmetry φ→ φ+ δφ is

δS[φ] ≡ S[φ+ δφ]− S[φ] =

∫ddx ∂µKµ. (1)

I An arbitrary variation of the action takes the form

δS =

∫ddx[−E(φ)δφ+ ∂µθ

µ(φ;δφ)], (2)

where E(φ) are the equations of motion and θµ is the symplectic current density.I Consider a deformation of the symmetry by a local function ρ

δρφ = ρδφ.

Assuming S depends only on φ and ∂µφ, (1) is modified to

δρS =

∫ddx(ρ∂µKµ + (∂µρ)θµ(φ;δφ)

),

and evaluating (2) on δρφ gives

δS∣∣∣δρφ

=

∫ddx[−E(φ)ρδφ+ (∂µρ)θµ(φ;δφ) + ρ∂µθ

µ(φ;δφ)].

[Noether (1918); S. Avery & B. Schwab, hep-th/1510.07038]Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 4 / 28

Noether’s First Theorem (continued)

I Equating the following

δρS =

∫ddx(

(∂µρ)θµ(φ;δφ) + ρ∂µKµ),

δS∣∣∣δρφ

=

∫ddx[−E(φ)ρδφ+ (∂µρ)θµ(φ;δφ) + ρ∂µθ

µ(φ;δφ)],

one finds

0 =

∫ddxρ(x)

[−E(φ)δφ+ ∂µ

(θµ(φ;δφ)− Kµ

)].

Since ρ is arbitrary, the integrand must vanish.

Noether’s First Theorem:The current jµ of a continuous symmetry is conserved on the equations of motion

∂µjµ = E(φ)δφ ≈ 0, where jµ = θµ(φ;δφ)− Kµ.

? Note, the derivation did not depend on form of δφ (other than being infinitesimal),so it applies just as well to local symmetries.


Noether’s Second Theorem

Noether’s Second Theorem:The Noether current jµ of a local symmetry can always be written as

jµ = Sµ + ∂νkνµ,

where Sµ ≈ 0 and kµν = −kνµ.

I Consider local symmetries, parametrized by an arbitrary function ε(x).For simplicity, consider symmetries of the form

δεφ = f (φ)ε+ f µ(φ)∂µε.

I To derive an identity involving the equations of motion (which will beindependent of ε), focus on ε of compact support.

I Again, compare (1) with (2) evaluated on δεφ,(total derivatives will not contribute)∫

ddx E(φ)δεφ = 0.

[Noether (1918); S. Avery & B. Schwab, hep-th/1510.07038]


Noether’s Second Theorem (continued)

I Since ε is compactly supported, integrate-by-parts to find∫ddx ε ∆(E) = 0, ∆(·) ≡ f (φ)(·)− ∂µ(f µ(φ) ·).

I Then, since ε is arbitrary, this implies

∆(E) = 0.

I Comments:? Identity involving only the equations of motion.? Implies equations of motion are not all independent⇒ some degrees of freedom are gauge.

I Takeaway:? Theories with local symmetries of compact support are gauge theories.

[Noether (1918); S. Avery & B. Schwab, hep-th/1510.07038]


Noether’s Second Theorem (continued)

I Now use identity to derive decomposition of the Noether current. Recall,∫ddx E(φ)δεφ

∫by parts=⇒

∫ddx ε ∆(E).

I This implies the integrands differ by a total derivative

Eδεφ = ε∆(E) + ∂µSµ(E;ε).

I However, since ∆(E) = 0, this implies

∂µSµ(E;ε) = Eδεφ.

Comments:? Sµ obeys the same conservation equation as the Noether current jµ.? Sµ vanishes on the equations of motion.

I Then, taking the difference

∂µ(jµ(ε)− Sµ(ε)) = 0,

implies (assuming trivial de Rham cohomology)

jµ(ε) = Sµ(ε) + ∂νkνµ(ε), kµν = −kνµ.


Conserved Charges in Gauge TheoryI Decomposition reveals that we can construct conserved charges from kµν

Qε =

∫σ=∂Σ

dσµνkµν(ε) ≈∫

Σ

dΣµjµ(ε).

I Conservation of charge is only non-trivial if ε has support at the boundary ∂Σof the Cauchy surface. Otherwise, charge is zero and trivially conserved.

I Example: QED

e2jµ = εe2jµM − (∂νε)Fµν , e2Sµ = ε(

e2jµM + ∂νFµν), e2kµν = εFµν .

Qε =1e2

∫σ

dσµν εFµν

I Example: Non-abelian gauge theory

g2jµ = Tr[εg2jµM − (∂νε− i[Aν ,ε])Fµν

],

g2Sµ = Tr[ε(

g2jµM + ∂νFµν − i[Aν ,Fµν ])], g2kµν = Tr[εFµν ].

Qε =1g2

∫σ

dσµν Tr[εFµν ]

? Recover standard global charge when ε is constant.Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 9 / 28

Conservation in Minkowski Space

I Use conformal diagram to studyasymptotic boundary of flat space.

I Diagram preserves causal structure.

lightray

lightray

mas

sive

part

icle

I+

i0

I+

I+r=1

i+ I++

r=1

r=1

iI

Ir=1

const rconst t

1

I I± are natural Cauchy surfaces formassless scattering.

I Charges of local symmetries are

Q±ε =

∫I±∓

dσµνkµν(ε).

I±∓ are boundaries of I± near i0.I One version of conservation is

Q+ε = Q−ε .

I Instead we’ll focus on conservationmeasured by observers on I+.(These are natural observers ofscattering in the bulk.)


Conservation for Observers at I+

ui

uf

I+r=∞

r =∞

x(z,z)

x(z,z)

I Coordinates for observers at I+:

u = t − r, r2 = ~x2, ~x = rx(z,z)

? I+: r →∞ at fixed u? u is observer time on I+

? (z,z) are coordinates on S2

with covariant derivative D

I Charge:Q+ε =

1g2

∫I+−

d2x Tr[εFru]

I Study change in charge as a function of observer time u∫S2

d2x Tr[εFru]∣∣∣

uf

−∫

S2d2x Tr[εFru]

∣∣∣ui

=

∫S2

d2x∫ uf

ui

du Tr[DzεFzu + DzεFzu − εJu

],

whereJu ≡ i[Az,Fzu] + i[Az,Fzu] + g2jM

u .

I Since ε is arbitrary, obtain local (on S2) conservation law

∆Fru = −∫ uf

ui

du[DzFzu + DzFzu + Ju

].

I These are the asymptotic symmetries of interest.I They are physical because they imply a non-trivial conservation law.


Key Points

1. Generic gauge theories enjoy infinite-dimensional physical symmetries,known as asymptotic or large gauge symmetries.? Noether’s 2nd theorem⇒ physical symmetries are non-vanishing at the

boundary.? They are physical because they imply a non-trivial conservation law.? We will focus on non-trivial conservation law that arises at null infinity

(I).2. The memory effect is the observable consequence of large gauge

symmetry.? The memory effect is the effect of a vacuum transition on a pair of probes

charged under the gauge group.




The Color Memory EffectI Consider scenario w/ vacuum transition

? Radiative vacuum before ui and after uf

Fuz = Fzz = 0.

⇒ Az = iU∂zU−1

? Color flux at intermediate times

ui

uf

I+r=∞

fluxco

lor

CS2: ds2=r2γabdzadzb

I In temporal gauge (Au = 0),rearrange conservation law

(Dz∆Az+Dz∆Az)

= ∆Fru −∫ uf

ui

du Ju.

I Find vacuum transition as afunction of color flux.(“memory”)

I Claim: Vacuum transitioninduces permanent rotation inrelative colors of ‘test’quarks.

[MP, A. Raclariu, & A. Strominger, hep-th/1707.08016]


The Color Memory Effect (continued)

I Claim: Vacuum transitioninduces permanent rotation inrelative colors of ‘test’ quarks.

I Quarks begin in singlet at ui

I At uf , quarks have acquiredrelative color rotation whosetrace is

WC ≡1

NcTr Pexp

(i∮CA)

where C is closed contour onI+.⇒ measures vacuum transition

ui

uf

I+r=∞

z1 fluxco

lor

z2A = 0

A 6=0

C

[MP, A. Raclariu, & A. Strominger, hep-th/1707.08016]


Key Points


2. The memory effect is the observable consequence of large gaugesymmetry.? The color memory effect is a permanent relative color rotation of a pair of

“test” quarks induced by the transit of color flux across null infinity.? The color memory effect is a measure of the vacuum transition.




Color Memory in the Regge Limit

I The Regge limit is a limit of fixed momentum transfer, with center-of-massenergy taken to∞

t fixed, s→∞.I In deeply inelastic scattering, we find

xBj ≡ −q2

2P · q ∼ts→ 0

where q is momentum of exchanged photon and P is the hadron momentum.I Since xBj is fixed by kinematics to be the longitudinal momentum fraction

carried by the struck parton, the Regge limit probes partons (gluons) carrying asmall fraction x of the hadron momentum.

I Introduce lightcone coordinates

x± =t ± x3√

2, ~x = (x1,x2).

I To resolve dynamics at small-x, work in infinite momentum frame (IMF).(P+ →∞ for hadron moving in x+ direction).


Color Memory in the Regge Limit (continued)

I To see why IMF resolves dynamics at small-x, notice typical lifetime of aparton with lightcone momentum k+ = xP+

∆x+ ∼ 1k−

=2k+

m2⊥

= x2P+

m2⊥

I Hence, we can use a Born-Oppenheimer type approximation and treat large-xd.o.f. as static sources for gluons at small-x.

I Next from longitudinal spread

∆x− ∼ 1k+

=1

xP+

find large-x d.o.f. are highly localized in x−.I For purposes of small-x dynamics, fixing A+ = 0 gauge, we can approximate

large-x d.o.f. by a color shockwave traveling in the x+ direction

g2JµM = δµ+δ(x−)ρ(~x).

? Resembles localized color flux through I+ that induces vacuum transition.


Color Memory in the Regge Limit (continued)

I Taking static field configurations, A− = 0 and no long. mag. fields (Fij = 0),and integrating the only non-trivial component of the YM equations, one finds

−∂i∆Ai =

∫ x−f

x−i

J−, J− = −δ(x−)ρ(~x)− i[Ai,∂−Ai].

I Resembles vacuum transition memory formula, but to make precise, must placeat I+.

I First must identify analogue of IMF for I+ observer? LC coordinates are nice coordinates for the IMF because they transform

simply under boosts in x3 direction

(x+,x−)→ (λx+,λ−1x−).

? The IMF is reached by taking λ→∞. In LC coordinates, can readilyobtain IMF configurations from configurations in other inertial frames.

[A. Ball, MP, A. Raclariu, A. Strominger & R. Venugopalan, hep-th/1805.12224]


The Regge Limit and Null InfinityI (u,r,z,z) do not transform nicely (i.e. by scaling) under boosts in x3 direction.I Trick: obtain nice coordinates by singular coordinate transformation

(r,u,z,z)→ (λr,λ−1u,λ−1z,λ−1z), λ→∞.I In new coordinates, further rescalings are boosts in x3.I S2 is flattened to transverse plane.I Related to LC coordinates by

x+ =√

2r, x− =1√2

(u + rzz), x1 + ix2 = 2rz.

I Can identify r →∞ limit with x+ →∞ limit!I However, vacuum transition formula was x+-independent⇒ can place at I+ for free!

Small-x gluon field configurations are the vacuum-to-vacuum field configurationsgoverned by the conservation law for large gauge symmetry.

I Can we measure these?I In other words, what are the analogues of the “test” quarks?

[A. Ball, MP, A. Raclariu, A. Strominger & R. Venugopalan, hep-th/1805.12224]Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 19 / 28

Color Memory in Collider Observables

I Goal: seek observables sensitive to vacuum-to-vacuum configurations.I Consider electron-ion DIS.

e−

e−

γ∗

q

q

heavy ion

I Focus on process where virtual photonfluctuates into singlet quark-antiquark pair.

I In eikonal approximation, shockwave inducescolor rotation on each quark.

I Forward scattering amplitude:

S(~x1,~x2) =1

NcTr[U(~x1)U†(~x2)

]=WC .

Identify amplitude with quark dipole colorrotation!

[A. Ball, MP, A. Raclariu, A. Strominger & R. Venugopalan, hep-th/1805.12224]


Color Memory in Collider Observables (continued)

I To obtain observable, must average over color sources (CGC).? Large random rotations average to zero, small rotations approx. identity? Emergent scale: size of dipole where transition occurs.

I Dipole cross-section: (via optical theorem)

σdipole(x,~r) = 2∫

d2~b [1− 〈Re S(~x1,~x2)〉],

where~r = ~x1 −~x2 and ~b = (~x1 +~x2)/2.I Inclusive DIS virtual photon-heavy ion cross-section:

σγ∗ion(x,Q2) =

∫ 1

0dz∫

d2~r |Ψ(z,~r,Q2)|2γ∗→qqσdipole(x,~r),

where |Ψ(z,~r,Q2)|2γ∗→qq is probability for γ∗ → qq with dipole of size~rand quark carrying momentum fraction z.


Key Points




3. The color memory effect appears in scattering at collider energies in theRegge limit of QCD.? Vacuum-to-vacuum transitions governed by the large gauge conservation

law appear in the Regge limit.? Observables in the Regge limit are sensitive to these transitions.



Large Gauge Symmetry and Quantum Information

I Color rotation→ learn something about flux of color radiation through I+

I Imagine dense array of “test” quarks→ can fully determine ∆Az

I Ask: To what extent can we distinguish scattering events in the bulk?I Answered by Carney, Chaurette, Neuenfeld, & Semenoff

(hep-th/1706.03782, hep-th/1710.02531),(see also Strominger, hep-th/1706.07143)? Study scattering in QED and gravity? Determine reduced density matrix for scattered hard particles by tracing

over soft radiation? Find soft radiation decoheres nearly all momentum superpositions of

outgoing hard particles? Intuition: radiation is essentially classical and distinguishes different

scattering events


Large Gauge Symmetry and Quantum InformationI For simplicity, focus on QED.I In scattering, analogue of conservation law is the Ward identity

〈β|(Q+ε S − SQ−ε

)|α〉 = 0.

I As before, use equations of motion (Gauss constraint) to write as

Q+ε =

∫I+

dud2x εjMu −1e2

∫I+

dud2x(DzεFuz + DzεFuz

)≡ Q+

H + Q+S .

I Need similar expansion of charge Q−ε near I− to determine action on |α〉.I Rearrange and simplify

〈β|(Q+

S S − SQ−S)|α〉 = −〈β|

(Q+

HS − SQ−H)|α〉

= Ωε(β,α)〈β|S|α〉,where

Ωε(β,α) =1

4π

∫d2x ε D2Ωβα ∼

∫d2x εr2Er,

Ωβα(x) = −∑k∈β

Qklog(pk · q) +∑k∈α

Qklog(pk · q), qµ = (1, x).

[CCNS - hep-th/1706.03782, hep-th/1710.02531]Monica Pate Large Gauge Symmetry & Memory Effects September 12, 2018 24 / 28

Large Gauge Symmetry and Quantum Information

I Why soft? → Express in terms of standard creation and annihilation operators.

Q+S =

18πe

∫d2x[Dzε∂zxi + Dzε∂zxi

]limω→0

∑α=±

[ωεα∗i aα(ωx) + ωεαi a†α(ωx)

]I In quantum scattering, conservation law determines soft photon content.I Namely, suppose Q−S |α〉 = 0, then, from

〈β|(Q+

S S − SQ−S)|α〉 = Ωε(β,α)〈β|S|α〉,

we find

|β〉 = exp

[e∫ Λ

0

dωd2x2(2π)3

[DzΩβα∂zxi + DzΩβα∂zxi

]∑α=±

[εα∗i aα(ωx)− εαi a†α(ωx)

]]|β〉

≡ Wβα|β〉.

I These are a generalization of the Fadeev-Kulish states.

[CCNS - hep-th/1706.03782, hep-th/1710.02531][Fadeev & Kulish (1970), KPRS-hep-th/1705.04311]


Large Gauge Symmetry and Information Theory

I Consider evolution of density matrix associated to an incoming state

|α〉〈α| → S|α〉〈α|S† ≡ ρ.I Trace over soft radiation to compute reduced density matrix

ρred = Trsoft(ρ) =∑ββ′

〈0|W†β′αWβα|0〉 SβαS∗β′α|β〉〈β′|, Sβα = 〈β|S|α〉,

〈0|W†β′αWβα|0〉 =

(λIR

Λ

)Γββ′

=

1, Γββ′ = 00, Γββ′ > 0 ,

Γββ′ =e2

2

∫d2x

(2π)3 γzz∂z(Ωββ′)∂z(Ωββ′).

I α contribution cancels between two dressings and drops out.I Integrand of Γββ′ is the norm of a vector on S2 ⇒ Γββ′ ≥ 0.I Γββ′ = 0 when Ωββ′ is constant. (∼ radial electric fields match)I In QED, one finds ρred is diagonal, hence completely decohered.

[CCNS - hep-th/1706.03782, hep-th/1710.02531]


Key Points




3. The color memory effect appears in scattering at collider energies in theRegge limit of QCD.? Vacuum-to-vacuum transitions governed by the large gauge conservation

law appear in the Regge limit.? Observables in the Regge limit are sensitive to these transitions.

4. In scattering events, large gauge symmetry implies soft radiation (orvacuum transitions) is highly correlated with hard particles.? Soft radiation nearly completely decoheres outgoing momentum

superpositions.


Conclusions/Outlook

I Large gauge (asymptotic) symmetries are symmetries of generic gaugetheories with physical consequences.

I Color memory, the observable consequence of large gauge symmetry inYang-Mills theory, appears in scattering at collider energies in the Reggelimit of QCD.

I Appearance of asymptotic symmetries in Regge limit is intriguing anddeserves further investigation.

I Entanglement between degrees of freedom at small-x and degrees offreedom at large-x due to large gauge symmetry?


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